with \(\theta \ne n\pi \) , with \(n\)  an arbitrary integer. All other fields remain
invariant under this symmetry. As we see, the symmetry treats differently one of the
generations of quarks In fact, these are unrotated quark fields, not yet
corresponding to physical quarks, but the principle holds., since only the “first
family” of quarks is affected by the transformations above. In fact, there are six
(not counting the leptonic sector)
models of the BGL type, which depend on which generation of quarks is chosen in eq:sym above. For the scalar sector, the above symmetry transformation yields a Peccei-Quinn [1]} scalar potential, which must be complemented with a soft breaking parameter as to yield a massive pseudoscalar particle.
